---
tag: T9DG8DDE75
kind: concept
name: Pointed space
msc: [55P99, 55Q05]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Pointed_space
same_as:
  wikidata: Q1419329
  wikipedia: Pointed space
---

## Summary

A pointed space is a [[topological space|0CQHFPGJ45]] together with a chosen point, its base point.
Maps of pointed spaces are continuous maps that send base point to base point.
Many constructions of homotopy theory need a base point, such as the [[fundamental group|S1XK3HXHFX]], the [[loop space|2JFF5F3EGF]] and the [[wedge sum|5JSWM1HBX2]].

## Explanations

### L1 human

A pointed space is a space with one marked point, like a map of a city with a pin at your home.
Loops in a pointed space start and end at the pin.
For example, the circle with the marked point $1$ is a pointed space, written $(S^1, 1)$.
A map of pointed spaces must send the pin of one space to the pin of the other.